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The mean field spin glass model is analyzed by a combination ofmathematically rigororous methods and a powerful Ansatz. The method exploitedis general, and can be applied to others disordered mean field models such as,e.g., neural networks.  It is well known that the probability measure of overlaps among replicascarries the whole physical content of these models. A functional orderparameter of Parisi type is introduced by rigorous methods, according toprevious works by F. Guerra. By the Ansatz that the functional order parameteris the correct order parameter of the model, we explicitly find the fulloverlap distribution. The physical interpretation of the functional orderparameter is obtained, and ultrametricity of overlaps is derived as a naturalconsequence of a branching diffusion process.  It is shown by explicit construction that ultrametricity of the 3-replicasoverlap distribution together with the Ghirlanda-Guerra relations determinesthe distribution of overlaps among s replicas, for any s, in terms of theone-overlap distribution.Comment: 17 pages, submitted to Euro. Phys. Jou. B Direc

Some exact results on the ultrametric overlap distribution in mean field spin glass models (I)